Partially ordered set

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In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. This relation formalizes the intuitive concept of an ordering, sequencing, or arrangement of the set's elements. Such an ordering does not necessarily need to be total, that is, it need not guarantee the mutual comparability of all objects in the set, but it can be. (In mathematical usage, a total order is a kind of partial order.) A poset defines a poset topology.

1 Formal definition
2 Examples
3 Strict and weak partial orders
4 Linear extension
5 Category theory
6 Partial orders in topological spaces
7 References
8 See also

[edit] Formal definition

A partial order is a binary relation R over a set P which is reflexive, antisymmetric, and transitive, i.e., for all a, b, and c in P, we have that:

aRa (reflexivity);
if aRb and bRa then a = b (antisymmetry);
if aRb and bRc then aRc (transitivity).

A set with a partial order is called a partially ordered set (also called a poset). The term ordered set is sometimes also used for posets, as long as it is clear from the context that no other kinds of orders are meant. In particular, totally ordered sets can also be referred to as "ordered sets", especially in areas where these structures are more common than posets.

Often times one uses the more familiar "≤" notation to denote the partial order instead of "R".

[edit] Examples

Some of the principal examples are these:

The set of subsets of {x,y,z}, ordered by inclusion

The set of natural numbers equipped with its natural ordering (the less than or equal to relation). This partial ordering is a total ordering.

The set of integers equipped with its natural ordering. This partial ordering is also total.

A finite subset {1, 2, ..., n} of the set of natural numbers. This partial ordering is also total.

The set of natural numbers equipped with the relation of divisibility.

The set of subsets of a given set (its power set) ordered by inclusion.

The set of subspaces of a vector space ordered by inclusion.

[edit] Strict and weak partial orders

In some contexts, the partial order defined above is called a weak (or reflexive) partial order. In these contexts a strict (or irreflexive) partial order is a binary relation that is irreflexive and transitive, and therefore asymmetric. In other words, for all a, b, and c in P, we have that:

¬(aRa) (irreflexivity);
if a ≠ b and aRb then ¬(bRa) (asymmetry); and
if aRb and bRc then aRc (transitivity).

If R is a weak partial order, then R − {(a, a) | a in P} is the corresponding strict partial order. Similarly, every strict partial order has a corresponding weak partial order, and so the definition of each is readily expressed in terms of the other.

Strict partial orders are also useful because they correspond more directly to directed acyclic graphs (dags): every strict partial order is a dag, and the transitive closure of a dag is both a strict partial order and also a dag itself.

[edit] Linear extension

A total order T is a linear extension of a partial order P if, whenever x ≤ y in P it also holds that x ≤ y in T. In computer science, algorithms for finding linear extensions of partial orders are called topological sorting.

[edit] Category theory

When considered as a category where hom(x, y) = {(x, y) | x ≤ y} and (y, z)o(x, y) = (x, z), posets are equivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if any, is an initial object, and the largest element, if any, a terminal object. Also, every pre-ordered set is equivalent to a poset. Finally, every subcategory of a poset is isomorphism-closed.

[edit] Partial orders in topological spaces

If P is a topological space, then it is customary to assume that R is closed in $P\times P$ . Under this assumption relations are well behaved in limits; if $a_i\to a$ and $a i R b$ for all i, then $a R b$ . See Deshpande (1968).